I understand you want to implement a Maximum Likelihood detector in MATLAB. To perform an exhaustive search over “Xgssk” and decode “Y”, you can follow these steps:
1. Equalize "Y" so that it has the same dimension as "X" by applying the pseudo-inverse of the channel matrix "H". You can do this using the pinv function in MATLAB as follows:
H_pseudo_inv = pinv(H); % Compute pseudo-inverse of H
Y_hat = H_pseudo_inv * Y; % Recovered signal
Refer to the following documentation link for more information on pinv function:
2. Loop over all the columns in the “Y_hat” and decode each column to the closest column in “Xgssk” based on Euclidean distance.
decoded_indices = zeros(1, L); % Store the decoded indices
for l = 1:L
y_col = Y_hat(:, l); % Extract current column
distances = vecnorm(Xgssk - y_col, 2, 1); % Euclidean distance
[~, closest_idx] = min(distances); % Find index of closest column
decoded_indices(l) = closest_idx; % Store the index
end
In this way, you can obtain the indices of the corresponding columns in “Xgssk” for each column in “Y” allowing you to recover the maximum Likelihood estimate of "X".
I hope this helps!
